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\title{Propaganda to a Cynical Audience\thanks{The author thanks Scott Gehlbach, Sergei Guriev, Zhaotian Luo, Adam Przeworski, Anton Shirikov, and Konstantin Sonin for their comments and feedback.}}
\author{Alexei Zakharov\thanks{al.v.zakharov@gmail.com. Associate Instructional Professor, The University of Chicago}}

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\begin{abstract}
Using a model, we explain why propaganda in autocracies can be blatantly false and unconvincing. We model two news outlets that report on a hidden state of the world, motivated by the ex-post beliefs of the audience about the state of the world. News outlets face a tradeoff when making egregiously false statements. On the one hand, such statements are easily verifiable as false. On the other hand, a demonstrably false report reduces the credibility of the report made by the competing outlet. This is especially true for audiences in autocracies that are characterized by high media cynicism and are prone to making sweeping generalizations about the self-serving nature of all media. 
\end{abstract}

\textit{ Keywords: }Propaganda, cynicism, media, autocracy

\textit{ JEL codes:} D03, D72, D82, D83

\begin{center}
\textit{Conditionally accepted manuscript, Political Science Research and Methods}
\end{center}
\vfill

\clearpage 
\begin{quote}
...a despot easily forgives his subjects for not loving him, provided they do not love each other. \textit{Alexis De Tocqueville ``Democracy in America'' \citep{Tocqueville2000}}.
\end{quote}


\section{Introduction}

Propaganda in autocracies can be crude and unconvincing, making claims that are often egregious, conspicuously false, and mutually exclusive \citep{paul2016russian}, contrary to the more conventional understanding of propaganda as persuasive, instilling pro-regime values, and indoctrinating the populace to achieve political legitimacy \citep{brady2009mass,schaar1981legitimacy}. False and easily disprovable claims are also increasingly a feature of democratic politics, promoted by both politicians and media platforms \citep{allcott2017social,nyhan2020facts,jerit2020political,starbird2017examining}.

In this paper, we use a formal model to argue that crude and unconvincing propaganda can be effective when it exploits the audience's inherent media cynicism. Cynicism is the negative view that the actions of others are motivated by self-interest \citep{neumann2023toward}, as opposed to expressive motives such as values or integrity. Media cynicism, in particular, involves the uncritical attribution of self-serving motives to journalists and media outlets, and the 
rejection of the idea that values and professionalism could be a motive behind reporting \citep{markov2022understanding}. Media cynicism has recently been characterized as a way in which audiences relate to news media \citep{markov2023unpacking,tsfati2025media} that is distinct from skepticism or distrust. A distrustful skeptic deliberates over the motivations and professionalism of a news source to allay (or prove) his suspicions that the source is self-serving. A cynic makes negative moral judgments \citep{neumann2023toward}, resulting in sweeping, deterministic conclusions about the self-serving and instrumental motives of all media. 

Our paper argues that cynicism primarily matters in situations where the audience is exposed to sharply different reports by competing media outlets. In such cases, a less cynical individual will reflect on the motivations of different media sources and put more weight on reports that appear more truthful. For a cynic, a single untruthful-looking report will cast a shadow over news reporting as such, making him less likely to update on prior beliefs when witnessing sharply divergent reports, some of which appear truthful and others do not.\footnotemark{}
\footnotetext{Cynics generally resist attempts to change their opinions \citep{alyukov2023harnessing}; widespread political cynicism aligns with findings that people often recognize false news as false but misjudge true news as false \citep{shirikov2024fake,pfander2025spotting}.}
With a cynical audience, extremely exaggerated and untruthful news reporting will become more common, as it has the added benefit of making the reporting of outlets with competing objectives less credible.


Our analysis is premised on the assumption that some audiences are inherently more cynical or disposed to ignore the possibility of idealistic, ethical, or professional motives, and are more likely to think that individuals and organizations pursue purely instrumental and material goals \citep{citrin2018political,roudakova2017losing}. Audiences rely on prior levels of cynicism when interpreting political information \citep{dancey2012consequences}, leading to spillover effects when negative information about some politicians or institutions impacts others \citep{lee2018spillover}. Cynical beliefs are commonly formed under authoritarian practices \citep{walker2014breaking,zhelnina2020apathy,shields2021killing} or in a poor institutional environment \citep{stavrova2019cynical}.\footnotemark{} This way, we argue that crude and unconvincing propaganda can be effective in modern authoritarian regimes that are commonly characterized by political demobilization \citep{linz2000totalitarian} and the accompanying deep distrust of the media \citep{tsfati2014individual}, exploiting the audience's inherent cynicism to discredit alternative news sources.% We do not analyze why some political regimes decide to have a more cynical public; Instead, we focus on how the public's innate cynicism affects the quality of reporting by media outlets. 
\footnotetext{Specific styles of news coverage \citep{cappella1997spiral,jones2021perceptions} or exposure to conspiracy narratives \citep{jolley2014social,invernizzi2023trust} can increase mistrust in the political system and the perception that politicians are self-serving.}

Importantly, people are more likely to make sweeping, cynical generalizations about the corrupt and self-serving nature of media outlets after witnessing malfeasance, such as conspicuously false or biased news reporting. Experimental studies note that exposure to false news triggers a mindset where all news is distrusted \citep{altay2024exposure}. A similar pattern is present in observational studies \citep{ognyanova2020misinformation}, with spillovers of mistrust across different media channels \citep{lee2023antecedents}. \cite{mckay2021disinformation} argue that disinformation campaigns promote the distrust of media and objective knowledge acquisition as such. Because of this, cynicism has been characterized as a cognitive schema or a mental shortcut that helps an individual organize, quickly interpret, and store political information \citep{bartlett1995remembering,shields2021killing}. In response to conspicuously false or biased reporting, the cynical view of the media (and politics in general) is brought into awareness, making one more likely to draw sweeping conclusions attributing instrumental motives to all news, and to believe that professional-looking reporting merely masks self-interest \citep{cappella1997spiral}.%\footnotemark{}
%\footnotetext{Psychologists call this the activation of a cognitive schema --- stored knowledge or expectations triggered by environmental cues that influence thought or behavior.}


We formalize the argument assuming two news outlets, controlled by the state and the opposition, that observe the ruler's strength and inform the public about their observation. News outlets can be principled (invariably reporting what they observe) or self-serving, which can make false reports and maximize (or minimize, for the opposition outlet) the public's Bayesian expectation of regime strength. A false report may be perceived as unconvincing, revealing the outlet as self-serving. Based on the observed reports and their apparent truthfulness, the public infers both the ruler's strength and the motivations of news outlets.

The public's degree of cynicism is operationalized by the correlation between the types of the two news outlets. This assumption captures the essential characteristics of media cynicism and its impact on how people perceive and process news. In particular, this setting is strategically equivalent to the one where the types of the media outlets are uncorrelated, but the public and the media outlets themselves perceive the correlation as positive. Higher correlations correspond to more cynical audiences: If one observes a false report from one outlet (learning that the outlet is self-serving), the individual begins to view all media with suspicion and assigns a higher probability to the event that the other outlet is also self-serving. For a non-cynical public, the correlation will be zero --- learning the type of one news outlet will not, by itself, tell the public anything about the type of the other outlet. 

We find that the public's inherent cynicism increases news outlets' incentives to make reports that are egregiously false and bend the truth to a great degree. In the eyes of a cynical audience, getting caught making a false report discredits the competing news source. Suppose there are three levels of ruler strength: Weak, Normal, and Strong. The state news outlet reports the ruler is Strong, and the opposition outlet reports the ruler is Weak. In addition, the state outlet's report sounds false and unconvincing, while the other report looks credible. When inferring the ruler's strength, the public must consider two possibilities. First, the ruler can be Weak, with the state outlet greatly exaggerating the ruler's strength. Second, the ruler can be Normal, with both outlets making false reports, but only the state outlet getting caught. A more cynical public will assign greater probability to the second event. Thus, cynicism increases the incentive to make egregiously false reports --- as long as an unconvincing report does not reveal any additional information about the state of the world. A key insight from our analysis is that, with a cynical audience, the risk of appearing unconvincing and untruthful does not deter news outlets from making egregiously false reports. That can happen even if egregiously false reports are sure to sound unconvincing. This way, our work resolves an empirical puzzle in authoritarian politics: the use of propaganda that is easily verifiable as false. 

We describe other conditions leading to egregiously and demonstrably false reporting. First, uncertainty about the underlying state of the world should not be too great. This way, if news outlets make radically divergent reports and one of the reports sounds untrue, it is more likely due to both of them making moderately false reports rather than one of them reporting the truth and the other making a blatantly false statement. Second, the public should be confident in its ability to identify moderately false statements as false; otherwise, the public will identify any conspicuously false report as egregiously false. Finally, if the state is ex ante more likely to be strong than weak, the state media outlet will be more inclined to make egregiously false reports than the opposition outlet. 

The rest of this paper is structured as follows. Section \ref{section_example} describes a case study. Section \ref{section_literature} reviews relevant literature. Section \ref{section_model} formulates and analyzes the model. Section \ref{section_conclusion} concludes.
 
\section{Case study: Russian propaganda following the MH17 shootdown}
\label{section_example}
Russian state propaganda after the shootdown of Malaysian Airlines Flight 17 is an example of how crude, unconvincing, and inconsistent reporting has been used to sow doubt and confusion, undermining public trust in the media \citep{paul2016russian}. In June 2014, pro-Russian rebels shot down the passenger aircraft over eastern Ukraine with a missile system that was supplied by the Russian state. Within hours of the disaster, Russian state media engaged in a chaotic and contradictory media campaign, amplifying multiple theories to confuse public perception. Russian officials and state media initially suggested that a Ukrainian fighter jet shot down MH17. State television broadcasts showed alleged evidence -- crudely superimposed out-of-scale photos of a fighter jet and a passenger airplane on satellite imagery; this image was quickly exposed as fake \citep{shevchenko2014web}. The Russian Defense Ministry forwarded a different (and mutually exclusive) version involving a Ukrainian surface-to-air missile launcher \citep{kelley2014russia}. Other, even less credible theories emerged, such as that the plane was already filled with dead bodies before takeoff, suggesting a bizarre Western plot to discredit Russia. This type of coverage is characteristic of the Russian state media's lack of commitment to consistency and its willingness to quickly discard discredited narratives in favor of new ones \citep{paul2016russian}. It differs from more traditional approaches that emphasize credibility and the avoidance of contradiction.

Consumers of news on the Russian state television recognize that reporting there is manipulative and untrue. In a qualitative study \citep{shields2021killing}, interviewed subjects were acutely aware that news broadcasts and political talk shows from Russian state television were biased and intended to ``brainwash'' the viewers, and often expressed disgust after watching the clips presented to them. At the same time, these experiences reinforced preexisting cynical attitudes toward politics and the general misbelief that genuine democracy and faithful news reporting are not possible anywhere. This way, rather than turning people into supporters, propaganda reduced dissent by sustaining mistrust and political disengagement \citep{shields2021killing}.  


\section{Related literature}
\label{section_literature}

Unconvincing propaganda in authoritarian settings is argued to inform the populace that the regime is strong \citep{huang2015propaganda,hassan2022political}, including by affecting second-order beliefs \citep{huang2021propaganda}, and ultimately reducing the public's willingness to resist the regime \citep{huang2018pathology}. We argue that similar empirical patterns can result from a different mechanism. If the audience is cynical, hard propaganda
can help maintain regime stability by reducing the credibility of
independent news sources, potentially lowering the perceived value of anti-regime collective action.

%My account differs from other models studying why crude authoritarian propaganda can be directed to a sophisticated audience. 
\cite{little2017propaganda} assumes that some people are credulous and perceive propaganda at face value. Others are sophisticated and can discern it as false, but are still affected because collective behavior requires the cooperation of the credulous part of the population. In our setting, the sophistication of the public is homogeneous, and we explicitly model the mechanism through which cynical but otherwise Bayesian rational individuals come to stop trusting the news. 

Other studies examine the impact of media outlets with varying objectives on accountability and welfare. \cite{li2022propaganda} argue that biased alternative media can contribute to democratic breakdown, enabling propaganda outlets to persuade voters to support an aspiring autocrat. In \cite{wolton2019biased}, a voter updates information on the state of the world and leader type from sources with known reporting strategies. Competing news sources can interfere with each other's messages, preventing the public from learning the truth \citep{minozzi2011jamming}. In contrast, concerns about news sources' reputations can prevent truthful reporting \citep{sheen2021media}. Our study differs from those in several essential respects. We assume that the types of news outlets are unknown to the public but correlated, so, as the public updates beliefs on both the state of the world and the types of news outlets, making visibly false and exaggerated reports can be an attractive strategy. 

More broadly, our work studies how autocracies manipulate public beliefs to reduce dissent. The strategy we describe is distinct from capturing, censoring or suppressing alternative news sources \citep{louis2023silence, shadmehr2015state,kolotilin2022censorship,glassel2020sometimes} or inducing self-censorship through the fear of reprisals \citep{gehlbach2022model}. 
\cite{kang2024making} assume that the incumbent can jam undesired signals by directly manipulating the perceived probability that a piece of news is false; we show that it can be achieved through reporting on the state of the world. In an intriguing model, \cite{sonin2025reverse} shows how an autocratic ruler can avoid accountability by lying about his performance and thus reducing the credibility of good performance by leaders in other countries. Our work is different in that it models two competing news outlets reporting on a domestic agenda, and an outlet can leverage media cynicism of the audience if the state of the world is unfavorable. Other works assume the possibility of voters being directly persuaded by propaganda \citep{horz2021propaganda,szeidl2024political}.


%Related studies focus on confirmation bias leading people to persistently accept verifiably false claims \citep{kasamatsu2022strategic}.





Our analysis can also explain why autocracies can allow some media freedom. Earlier explanations focus on aspects such as control over the bureaucracy \citep{egorov2009resource}, controlling corruption while managing social tensions \citep{lorentzen2014china}, avoiding incentives for censorship circumvention \citep{hobbs2018sudden}, or making independent media appear unattractive to regime supporters \citep{shirikovrethinking}. Our account is that the regime can utilize hard propaganda to exploit the public's predisposition to cynicism, thereby limiting the impact of alternative information sources on political attitudes and, ultimately, on collective action. The tradeoff between repression and propaganda, argued to be central to dictatorships \citep{guriev2019informational,guriev2020theory}, can also be similarly affected.  



\section{The model}
\label{section_model}
There are two media outlets, state $S$ and opposition $O$. Each outlet $i\in\{S,O\}$ can be either principled ($b_i=0$) or self-serving ($b_i=1$) with probability $\frac{1}{2}$ each. Let the correlation between the values $b_S$ and $b_O$ be $\rho\in(-1,1)$. 

We have $P(b_S=1|b_O=1)=\frac{1+\rho}{2}$ and $P(b_S=1|b_O=0)=\frac{1-\rho}{2}$. So, if $\rho>0$, then knowing that one of the sources is self-serving increases the observer's belief that the other source is self-serving. This is what should happen if the observer, after witnessing malfeasance from one news outlet, makes generalizations about the media as a whole. In the limiting case of $\rho=1$, observing a false report from one news source leads one to immediately reach the conclusion that faithful, professional reporting is impossible, and that the other news source is invariably self-serving. For values of $\rho$ closer to 0, learning the type of one news source, by itself, does not convey any information about the type of the other: people are willing to admit that news sources are principled or self-serving for purely idiosyncratic reasons. Finally, the polar case of $\rho=-1$ corresponds to the tendency to adopt a black-and-white worldview quickly: if one outlet is self-serving, the other must be principled. 

Suppose that the media outlets observe the level of ruler strength $x\in\{0,1,2\}$, where $x=0$ denotes a weak ruler, $x=1$ denotes a normal-strength ruler, and $x=2$ denotes a strong ruler. Ex-ante, the ruler is weak, normal-strength, or strong with positive probabilities $\frac{1-\beta}{2}$, $\beta$, and $\frac{1-\beta}{2}$, respectively, with $\beta\in(0,1)$ (later, one of our results will be obtained for generic positive probabilities over $\{0,1,2\}$). The media outlets then choose their reporting tactics. A principled outlet always reports the state of the world truthfully. A self-serving outlet $i\in\{O,S\}$ observing state $x$ can report any state $y_i\in\{0,1,2\}$. We will refer to a report $y\neq x$ as {\it false} and to a report $|y-x|=2$ as {\it egregiously false}. 

If a report $y$ by outlet $i$ is false, this becomes known to the outside observer with probability $r_{|x-y|}$, with $0<r_1\leq r_2=1$. In that case, the outside observer learns only that the report is false (and, hence, that the news source is self-serving) but not the actual state of the world $x$. We assume that an egregiously false report is also demonstrably false: if a news outlet makes an egregiously false claim, the public will know for sure that this claim is false, but may remain uncertain about how much the truth has been distorted.\footnotemark{}\footnotetext{In the Supplemental Appendix consider the more general case where $0<r_1\leq r_2\leq 1$.}

The {\it ex post} beliefs $p$ of the public about $x\in\{0,1,2\}$ are formed conditional on the reports $y_S$ and $y_O$ of the state and media outlets, and on whether any of them was observed making a false report. We do not explicitly model the public's actions after the ex post expectations are formed. Instead, it is assumed that both media outlets (if they are self-serving) value the public's ex-post beliefs about regime strength (and that the public is homogeneous), with the state-owned outlet seeking to portray the ruler as stronger and the opposition outlet aiming to portray the ruler as weaker. 

Without loss of generality, the public's belief that the state of the world is $x=0$ is valued by the state outlet with weight 0, the belief that $x=1$ is valued with a unit weight, and the belief that $x=2$ (or that the regime is strong) is valued with weight $\alpha>2$. The opposition outlet's payoffs are symmetric, with a weight of 0 on the belief that $x=2$, a weight of 1 on the belief that $x=1$, and a weight of $\alpha$ on the belief that $x=0$.  

Denote by $u_S(x,y_S)$ the expected payoff of $S$ if the state of the world is $x$ and report $y_S$ is chosen. When forming payoff expectation, $S$ must consider two possibilities. First, with some probability, $O$ is principled and, in that case, invariably chooses the truthful report $y_O=x$. Second, $O$ can be self-serving; in that case, the expectation is calculated over all possible values of $y_O$, considering the possibility that the falsehood of reports will become known to the public.\footnotemark{}
\footnotetext{See the proofs section for formal definitions of the payoff functions of the outlets.}

Denote by $q_{ix}(y)$ the probability that state $y$ is reported by outlet $i$ if the actual state is $x$ and the media outlet is self-serving, and by $q$ the profile of reporting strategies. We will use the following solution concept:
\begin{definition}
\rm
A {\it reporting equilibrium} will be reporting strategies $q$ and public beliefs $p$ such that 
\begin{enumerate}[label=\roman*)]
\item $q_{ix}(y)=0$ if $u_i(x,y)<\max_{y'}u_i(x,y')$,
\item In every information set visited with positive probability, $p$ are derived via Bayes rule from $q$; otherwise, $p$ must be the limit of a sequence of beliefs derived from a sequence of totally mixed strategy profiles converging to $q$,  
\item $q_{Sx}(y)=q_{O2-x}(2-y)$ for $x\in\{0,1,2\}$,
%\item $q_{Sx}(y)=0$ if $y<x$ and $q_{Ox}(y)=0$ if $y>x$,
\item $q_{Sx}(y)=0$ if $y<x$; $q_{Ox}(y)=0$ if $y>x$.
\end{enumerate}
\end{definition}
The first two conditions are standard for a sequential equilibrium in an extensive-form game \citep{kreps1982sequential}, applied to a game where observer beliefs enter directly into the utility functions of the players \citep{battigalli2009dynamic}. The third condition further requires any equilibrium to be symmetric. The fourth condition means no downward false reporting, when the state outlet underreports the ruler's strength or the opposition outlet overreports it.

%The observer's information set consists of 36 elements: $3\times 3=9$ report profiles times $2\times 2=4$ possible outcomes with respect to whether each outlet was revealed to be biased. For each outcome, we want to know the posterior probability for each state of the world $x\in\{0,1,2\}$. 

In an equilibrium, neither outlet will truthfully report its least desirable state of the world:

\begin{lemma}
\rm In any equilibrium we have $q_{S0}(0)=q_{O2}(2)=0$.
\label{lemma1}
\end{lemma}

Since we assume that a news outlet does not report a state that is less favorable than the true state, reporting one's least favorable state reveals it as the true state and results in zero payoff to the news outlet. At the same time, reporting one's most favorable state always yields a positive payoff. It follows that any equilibrium is characterized by a pair of probabilities $(q_1,q_2)$, where $q_1=q_{S1}(2)=q_{O1}(0)$ and $q_2=q_{S0}(2)=q_{O2}(0)$. Our next result is as follows: 

\begin{lemma}
    \rm
    There exists an equilibrium with $(q_1,q_2)=(0,1)$, no equilibria with $q_1=0,q_2<1$, and no equilibria with $q_2=0$. 
    \label{lemma_pathological}
\end{lemma}
This result implies that, while multiple equilibria are possible, we must have some egregiously false reporting. An equilibrium with no false reporting for the normal level of ruler strength requires very specific assumptions on the beliefs in an information set off the equilibrium path.\footnotemark{} 
\footnotetext{Choosing the report $y_S=2$ when $x=1$ can result in the public observing $y_S=2$, $y_O=1$, and outlet $S$ but not $O$ observed making a false report. If $q_1=0$, this information set lies off the game path, so the payoff of deviating to $y_S=2$ depends on what belief in this information set is assumed. Let $p_x$ be the probability assigned to state $x\in\{0,1,2\}$. Consider any sequence of mixed strategy profiles that converges to $q_1=0$, $q_2=1$, and does not allow downward reporting. While any belief such that $p_1+p_2=1$ can arise at the limit, only $p_1=1$ guarantees us that deviating to $y_S=2$ will not increase the payoff of $S$.}
In this equilibrium, both outlets make egregiously false reports with probability one. 

Our primary objective is to investigate the existence of equilibria where false reports are made with a positive probability in every state of the world, and to examine the comparative statics in the case where the equilibria are mixed-strategy. We have the following result.  %Our results, shown below, will qualitatively depend on whether the value of $\alpha$ is below 2, equal to 2, or above 2.

%\subsection{The case where $\alpha>2$.}

\begin{lemma}
\rm If $q_1>0$ in equilibrium, then we have $q_1=1$.
\label{lemma22}
\end{lemma}

%Only two types of equilibria may exist besides $q_1=0,q_2=1$: The pure-strategy equilibrium $q_1=q_2=1$ and a mixed-strategy equilibrium when an egregiously false report is made with probability less than one, and both outlets report falsely when the ruler is at normal strength. %In the special case $\alpha=2$, a continuum of equilibria, characterized by $q_2=1$ and different levels of $q_1$ may be possible; these equilibria are described in the Online Appendix.
Unless $q_1=0$, we are left with equilibria where a false report is always made if the ruler is at normal strength, and an egregiously false report is made with a positive (perhaps unit) probability. Such an equilibrium is unique and characterized by the following statement: 

\begin{proposition}
\rm In any equilibrium such that $q_1=1$, we have 
\begin{equation}
q_2=\min\left\{1,
\frac{r_1(1+\rho)(1-\beta+\beta r_1-\beta r_1\rho)}{(1-\beta)(1+r_1-\rho+r_1\rho)}\right\}.
\label{q2eq}
\end{equation}
\label{propq2}
\end{proposition}

The primary goal of our analysis is to study how the public's cynicism, characterized by parameter $\rho$, affects the probability that an egregiously false report is made in this equilibrium if the state of the world is unfavorable to that outlet (such as $x=0$ and the outlet is state-owned). Proposition \ref{propq2} tells us that, for a range of parameter values, the equilibrium is pure-strategy, meaning that an egregiously false report is always made, even though it is guaranteed to be recognized as false. This can occur if three conditions are met --- $\beta$ is sufficiently large, $r_1$ is sufficiently large relative to $\beta$, and the cynicism parameter $\rho$ is sufficiently large relative to $\beta$ and $r_1$: 

\begin{corollary}
\rm We have $q_2=1$ if and only if the following three conditions are satisfied: $\beta>\frac{1}{3}$, $r_1> \sqrt{\frac{1-\beta}{2\beta}}$, and $\rho\geq \frac{1-\beta}{\beta r_1^2}-1$. 
\label{corq2}
\end{corollary}

We now provide an intuitive explanation for this result. Suppose the state of the world is $x=0$. Then, the opposition outlet invariably reports $y_O=0$. Let the state outlet report $y_S=2$ and assume the report is perceived to be false. From the observer's position, this can happen under two circumstances. First, the ruler can be weak, with the state outlet misrepresenting the truth a great deal and the opposition outlet reporting truthfully. Second, the ruler can be at normal strength, with both outlets misrepresenting the truth to some extent, but only the state outlet getting caught. This means that making an egregiously false report always has some value, because even though the public will know the report is a lie, it will still be left guessing about the true state of the world. 

The payoff of reporting $y_S=2$ depends on the probabilities that the observer assigns to these two events, conditional on observing $y_S=2$, $y_O=0$, and on the state outlet's report being known to be false. The second probability is larger (and the first one is smaller) if the audience is cynical and believes that if one news outlet is self-serving, so is the other. For this logic to be valid, however, both $\beta$ and $r_1$ must be large enough so that the state of the world $x=1$ is plausible if $y_S=2$, $y_O=0$, and $S$'s report is observed as false. 

We proceed by looking at the comparative statics of $q_2$ in case the equilibrium is mixed-strategy. The following is true:

\begin{corollary}
\rm Let $q_2<1$. Then $\frac{\partial q_2}{\partial \rho}>0$ and $\frac{\partial q_2}{\partial \beta}>0$. Moreover, $\frac{\partial^2 q_2}{\partial\rho\partial\beta}>(<,=)\,0$ if $r_1<(>,=)\,\left(\frac{1-\rho}{1+\rho}\right)^2$.
\label{cor22}
\end{corollary}

As the audience becomes more cynical, the probability of egregiously false reports $q_2$ increases. %Values of $\rho$ closer to -1 correspond to an audience that readily identifies itself with XXXXXXX
For values of $\rho$ close to -1, egregiously false reports will be made rarely because being caught lying will increase the credibility of the opponent's report. An exception is when the probability that the ruler is at a normal strength is high --- in that case, two wildly divergent reports are both likely to be false.

As the extreme states of the world become less likely, the probability $q_2$ increases as well. To see this, suppose that the outlets report $y_S=2$, $y_O=0$, and that the state outlet's report is known to be false. This can potentially be a result of either $x=0$ or $x=1$ and both outlets making moderately false reports. The second is more likely if $\beta$ is large, so the effect of making an egregiously false report on the public's beliefs is larger if $\beta$ is large. 

Finally, Corollary \ref{cor22} states that if the probability that a moderately false report is exposed is small, then the effects of $\beta$ and $\rho$ discussed above are complementary: Blatantly false reporting is more sensitive to audience cynicism when the extreme states of the world become less likely.  

The next result investigates the comparative statics of $q_2$ with respect to $r_1$:
\begin{corollary}
    \rm Let $q_2<1$. Then $\frac{\partial q_2}{\partial r_1}>0$.
    \label{corr_33}
\end{corollary}

Potentially, $r_1$ has several different effects on equilibrium behavior. First, an increase in $r_1$ means that the observed outcome $y_S=2$, $y_O=0$, conditional on $S$ but not $O$ getting caught making a false report, is more likely to have resulted from $x=1$. This increases the attractiveness of choosing $y_S=2$ when $x=0$. For values of $r_1>\frac{1}{2}$, this effect reverses sign: if $r_1$ is close to 1 and $y_O=0$, it is unlikely that outlet $O$ will not be detected while making a false report. Second, a higher $r_1$ means a lower probability of $x=1$ (and a higher probability of $x=2$) conditional on observing $y_S=2$, $y_O=0$, and no visibly false reports; as $\alpha>2$, this results in a higher payoff from choosing the egregiously false report $y_S=0$ conditional on not getting caught. Third, as $r_1$ increases, the utility of reporting $y_S=1$ decreases. Hence, one may expect a nonmonotonic effect, but Corollary \ref{cor22} shows that the combined effect is always positive, and probability $q_2$ increases with $r_1$. 

In the final part of our analysis, we relax the assumption that states $x=0$ and $x=2$ are equally likely, allowing the ruler to be {\it ex ante} strong or weak. Denote by $\beta_0\neq\beta_2$ the probabilities of these two states. In such a case, Lemma 1 is still true, but there may not exist an equilibrium where the strategies of state and opposition outlets are symmetric. Therefore, the equilibrium strategy of the state outlet will be $q_{S1}$ or $q_{S2}$, or the probability that the report $y_S=2$ will be made if $x=1$ or $x=2$. Define $q_{O1}$, and $q_{O2}$ similarly. The following result is established:

\begin{proposition}
    \rm Let $\beta_0<\beta_2$. Then $q_{S2}\geq q_{O2}$ if $q_{S1}=q_{O1}=1$.
    \label{propnew}
\end{proposition}

Thus, if $\beta_0$ is small (so the ruler is {\it ex ante} strong), then state-owned media is more inclined to make egregiously false reports than the opposition media in any equilibrium where the outlets misreport if $x=1$. 


\section{Conclusion}
\label{section_conclusion}
Our work argues that crude, unconvincing propaganda will be effective because it discredits competing news sources even when the audience understands that the content of propaganda messages is being manipulated, provided that the audience is sufficiently cynical. 

We assume that the media outlets also believe, similarly to the audience, that there is a correlation between the types of outlets. Political and media elites themselves are often highly cynical, so the latter assumption is realistic.\footnotemark{} 
\footnotetext{For example, people in the position of power are more likely to attribute self-serving motives to generous acts \citep{inesi2012power}, while in corrupt nations clever people are more likely to be cynical \citep{stavrova2019cynical}.} More importantly, the equilibrium in this model does not differ from the one where the true correlation $\rho$ between the types of media outlets is known to the fully rational media outlets, and the public is rational except that it believes that the {\it ex ante} correlation between media outlet types is some $\hat\rho$ that is not necessarily equal to $\rho$.\footnotemark{}
\footnotetext{The cynical processing of news can be modeled more explicitly. For example, we can examine the model where the media types are uncorrelated, but a cynical response is triggered with probability $\lambda$ when an individual observes a false news report. The equilibrium will be similar to the one described in Proposition \ref{propq2}, with $\lambda$ playing the same role as $\rho$. This will happen for the same reason: A cynic is less likely to learn about the state of the world when media reports differ widely and one of the reports appears false.}

Our model can be extended in several ways. First, in the Supplemental Appendix we consider the case where the probability that an egregiously false report is exposed is less than one. Second, the current model assumes a homogeneous audience, with no social division with regard to the regime or the leader, and a uniform level of cynicism/sophistication. Future research can consider multiple audiences that vary in ex ante beliefs, or when a part of the public is loyal or gullible, perceiving messages from regime-affiliated news sources at face value and/or enjoying holding beliefs that are consistent with one's identity. Alternatively, general distrust in news sources can make one more likely to listen to messages that are congruent with one's position or identity. Third, it is for future research to study why authoritarian states not only benefit from a more cynical populace, but sometimes deliberately instill cynicism and mistrust \citep{alyukov2022propaganda,rosenfeld2024information}, benefiting from the political apathy of the populace \citep{gerschewski2023two,libman2024dictators}. A version of this logic was explored in \cite{brauninger2022political}, where competing elites can engage in a ``war on truth'', affecting the non-Bayesian public's perception of the quality of information that it receives from a news source; this can stop the voters from enacting policy
change that the elites currently oppose, at the expense of making them less likely to support the elites in the future. Further work can examine the different persuasive mechanisms that the state uses to influence the correlation parameter $\rho$, and how the decision to do so depends on the nature of threats faced by the regime: external, requiring the mobilization of regime supporters, or internal, requiring the suppression of anti-regime collective action. 

\clearpage

\section*{Appendix}
\subsection*{Definition of payoffs}


Denote by $d_i\in\{0,1\}$ whether it was revealed that outlet $i$ is self-serving, and let
$\eta_{d_i}(x,y_i)$ be the probability of $d_i$ given $x$ and $y_i$; so $\eta_0(x,x)=1$, $\eta_0(x,y)=1-r_{|x-y|}$ if $x\neq y$, and $\eta_1(x,y)=1-\eta_0(x,y)$. 

%The observer's information set consists of 36 elements: $3\times 3=9$ report profiles times $2\times 2=4$ possible outcomes with respect to whether each outlet was revealed to be opportunistic. 
Denote by $p:\{0,1,2\}^2\times\{0,1\}^2\rightarrow\Delta^2$ the belief of the public about state $x\in\{0,1,2\}$ following reports $y_S,y_O\in\{0,1,2\}$ and observations $d_S,d_O\in\{0,1\}$.\footnotemark{}
\footnotetext{The symbol $\Delta^2$ denotes the 2-dimensional simplex or the set of all probability distributions over $\{0,1,2\}$.}
Denote by $q_{ix}(y)$ the probability that state $y$ is reported by outlet $i$ if the actual state is $x$ and the media outlet is self-serving.

% Here we assume that $\alpha> 2$, and in the Appendix consider the case where $\alpha\in(1,2]$. 

Let $p_x(y_S,y_O,d_S,d_O)$ be the probability assigned to state $x\in\{0,1,2\}$ by the public depending on the reports made by both news outlets and whether these reports appear as false. The payoff of outlet $S$ if it chooses report $y_S\in\{0,1,2\}$ and the state of the world is $x$ is defined by taking the expectation of $p_1+\alpha p_2$ over all possible values of $y_O$, $d_S$, and $d_O$: 

\begin{align}
u_S(x,y_S)&=\frac{1-\rho}{2}\sum_{d_S}\eta_{d_S}(x,y_S)(p_1(y_S,x,d_S,0)+\alpha p_2(y_S,x,d_S,0))+\nonumber\\
&+\frac{1+\rho}{2}\sum_{y_O,d_S,d_O}q_{Ox}(y_O)\eta_{d_S}(x,y_S)\eta_{d_O}(x,y_O)(p_1(y_S,y_O,d_S,d_O)+\alpha p_2(y_S,y_O,d_S,d_O)).
\label{payoff_S}
\end{align}


The first summand of (\ref{payoff_S}) is the expected payoff of $S$ conditional on $O$ being principled and invariably choosing $y_O=x$, times $\frac{1-\rho}{2}$ or the probability that $O$ is principled if $S$ is self-serving. The second summand is the expected payoff of $S$ conditional on $O$ being self-serving (and, hence, choosing a $y_O$ that is potentially different from $x$), calculated over all possible realizations of $y_O$, $d_O$, and $d_S$ given $x$ and $y_S$ and multiplied by $\frac{1+\rho}{2}$ or the probability that $O$ is self-serving if $S$ is self-serving. 

Define the payoff of the opposition outlet symmetrically: 
\begin{align}
u_O(x,y_O)&=\frac{1-\rho}{2}\sum_{d_O}\eta_{d_O}(x,y_O)(p_1(x,y_O,0,d_O)+\alpha p_0(x,y_O,0,d_O))+\nonumber\\
&+\frac{1+\rho}{2}\sum_{y_S,d_S,d_O}q_{Sx}(y_S)\eta_{d_S}(x,y_S)\eta_{d_O}(x,y_O)(p_1(y_S,y_O,d_S,d_O)+\alpha p_0(y_S,y_O,d_S,d_O)).
\label{payoff_I}
\end{align}

\clearpage
\subsection*{Proofs of statements}

{\bf Proof of Lemma \ref{lemma1}.} Given any equilibrium strategies, we have $u_S(0,0)=0$. It follows that we should have $p_0=1$ if $(y_S,y_O,d_S,d_O)\in\{(1,0,0,0),(1,0,1,0),(2,0,1,0)\}$; otherwise, $S$ will not choose $y_S=0$ if $x=0$. As $p_0(1,0,0,0)=1$, it follows that we should have $q_{S1}(1)=0$ and $q_{S1}(2)=1$. This means that we observe $(y_S,y_O,d_S,d_O)=(2,0,1,0)$ with positive probability if $x=1$, which is a contradiction. {\bf Q.E.D.}\\


{\bf Proof of Lemma \ref{lemma_pathological}.} 
Here, we provide a sketch of the proof; for the full derivation, please refer to the Supplemental Appendix. We first derive $u_S(1,1)$, $u_S(1,2)$ by taking the expectations of posterior beliefs over the possible combinations of $(y_S,y_O,d_S,d_O)$. If $q_{S1}=q_{O1}=0$, then the information set $y_S=2$, $y_O=1$, $d_S=1$, $d_O=0$ is off the game path. If $q_{S2}=q_{O2}$, then for any sequence of totally mixed strategy profiles converging to $q_{S1}=q_{O1}=0$, $q_{S2}=q_{O2}<1$ we should have $u_S(1,2)>u_S(1,1)$, unless $q_{S2}=q_{O2}=1$. We then similarly derive $u_S(0,1)$ and $u_S(0,2)$; it immediately follows that $u_S(0,1)<u_S(0,2)$ if $q_{S1}=q_{O1}>0$ and $q_{S2}=q_{O2}=0$, ruling it out as an equilibrium. {\bf Q.E.D.}\\ 



{\bf Proof of Lemma \ref{lemma22}.} 
Assume $r_2\in(r_1,1]$, $q_{S1}=q_{O1}=q_1$, and $q_{S2}=q_{O2}=q_2$. Put
\begin{align}
&H_1(q_1,q_2)=u_S(1,2)-u_S(1,1)=\frac{(1-r_1)(1-\beta)(1-q_2)}{2}\frac{2(\alpha-1)+q_1(1+\rho)(2-\alpha)}{\beta q_1(2-q_1-q_1\rho)+(1-\beta)(1-q_2)}+\nonumber\\
+&\frac{1+\rho}{2}q_1(1-r_1)^2\frac{(\alpha-2) (1-\beta)q_2(1-r_2)}{\beta(1+\rho)q_1^2(1-r_1)^2+2(1-\beta)q_2(1-r_2)}+\nonumber\\
&+\frac{1+\rho}{2}q_1r_1(1-r_1)\frac{2\beta(1+\rho)q_1^2r_1(1-r_1)+\alpha(1-\beta)q_2r_2}{\beta(1+\rho)q_1^2r_1(1-r_1)+(1-\beta)q_2r_2},\\
&H_2(q_1,q_2)=u_S(0,2)-u_S(0,1)=\frac{\beta(1+\rho)q_1^2r_1(1-r_1)r_2}{\beta(1+\rho)q_1^2r_1(1-r_1)+(1-\beta)q_2r_2}+\nonumber\\
&+\frac{\beta(1+\rho)q_1^2(1-r_1)^2(1-r_2)+\alpha(1-\beta)q_2(1-r_2)^2}{\beta(1+\rho)q_1^2(1-r_1)^2+2(1-\beta)q_2(1-r_2)}-\frac{\beta q_1(1-r_1)(2-q_1-q_1\rho)}{\beta q_1(2-q_1-q_1\rho)+(1-\beta)(1-q_2)}.
\end{align}

For any equilibrium $(q_1,q_2)\neq(0,1)$, we must have $H_i=0$ if $q_i\in(0,1)$ or $H_i\geq 0$ if $q_i=1$, for $i=1,2$. If $\alpha>2$, then $H_1(q_1,q_2)>0$, so $q_1=1$. Now, $H_2(1,0)>0$, so for some $q_2\in(0,1]$ we have $H_2(1,q_2)=0$ and/or we have $H_2(1,1)>0$. {\bf Q.E.D.}\\ 

%If $q_2=1$ then $u_S(1,1)=1$ and $u_S(1,2)>(<,=)\:1$ if $\alpha>(<,=)\:2$. Hence, we cannot have $q_1=q_2=1$ in equilibrium if $\alpha<2$. 
%If $\alpha=2$ and $q_2=1$, then for any $q_1$ we have $u_S(0,1)>u_S(0,2)$, so we cannot have $q_1=q_2=1$ in equilibrium. 

{\bf Proof of Proposition \ref{propq2}.} Fix $r_2=1$. Then $H_1(q_1,q_2)>0$, so $q_1=1$. We have $u_S(0,1)$ increasing in $q_2$ and $u_S(0,2)$ is decreasing in $q_2$, with $u_S(0,1)<u_S(0,2)$ at $q_2=0$. So, we either have $q_2=1$, or a closed-form solution for $u_S(0,1)=u_S(0,2)$ in $q_2$ exists, given by the expression (\ref{q2eq}). 

It remains to show that backward misinformation is not optimal; please refer to Supplementary Appendix for details.\\  


{\bf Proof of Corollary \ref{corq2}.} We have $q_2=1$ if
$\beta(1+r_1^2(1+\rho))\geq 1$. The statement immediately follows. {\bf Q.E.D.}\\

{\bf Proof of Corollary \ref{cor22}.} Assume that $q_2<1$. Differentiating $q_2$ by $\beta$ and $r_1$ yields positive values. Now, $\frac{\partial q_2}{\partial \rho}$ has no more than one zero on $(-1,1)$, with $q_2=0$ for $\rho=-1$ and $q_2=1$ for $\rho=1$. This means that $q_2$ is strictly increasing in $\rho$ whenever $q_2<1$. {\bf Q.E.D.}\\


{\bf Proof of Proposition \ref{propnew}.} If $r_2=1$ we have $u_S(0,1)\leq u_S(0,2)$ if and only if 
$$
\frac{(2-q_{S1}-q_{S1}\rho)}{\beta_1 q_{O1}(2-q_{S1}-q_{S1}\rho)+2\beta_0(1-q_{S2})}\leq 
\frac{(1+\rho)q_{S1}r_1}{\beta_1(1+\rho)q_{S1}q_{O1}r_1(1-r_1)+2\beta_0q_{S2}}
$$
so
\begin{equation}
q_{S2}=\min\left\{1,\frac{(1+\rho)q_{S1}r_1\left(1+\frac{\beta_1}{2\beta_0}q_{O1}r_1(2-q_{S1}(1+\rho))\right)}{2-q_{S1}(1+\rho)(1-r_1)}\right\}.
\end{equation}
Similarly, we have 
\begin{equation}
q_{O2}=\min\left\{1,\frac{(1+\rho)q_{O1}r_1\left(1+\frac{\beta_1}{2\beta_2}q_{S1}r_1(2-q_{O1}(1+\rho))\right)}{2-q_{O1}(1+\rho)(1-r_1)}\right\}.
\end{equation}
The statement follows immediately for $q_{S1}=s_{O1}=1$.
{\bf Q.E.D.}\\


\clearpage  

\bibliographystyle{apsr}
\bibliography{aaa}

\clearpage



\appendix


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\begin{center}
    \textbf{Supplemental Appendix for\\
{\Large Propaganda to a Cynical Audience}
    }
\end{center}
\thispagestyle{empty}


\addcontentsline{toc}{section}{Appendix} % Add the appendix text to the document TOC
\part{} % Start the appendix part
\parttoc


\clearpage

\section{Proofs of Lemma \ref{lemma_pathological} and Proposition \ref{propq2}.}

{\bf Proof of Lemma \ref{lemma_pathological}.}
Assume $r_2\in(r_1,1]$ and consider the generic case where the strategies of media outlets are $y_{S1}$, $y_{S2}$, $y_{O1}$, and $y_{O2}$, denoting the probabilities of $y_S=2$ in states 1 and 0, and of $y_O=0$ in states 1 and 2. 

{\em Step 1.} Let $x=1$ and suppose $y_S=1$. Then the possible combinations of $(y_S,y_O,d_S,d_O)$ are given by $(1,1,0,0)$, $(1,0,0,0)$, and $(1,0,0,1)$. Conditional on $b_S=1$ and $x=1$, their probabilities are $\frac{2-q_{1O}-q_{1O}\rho}{2}$, $\frac{q_{1O}(1-r_1)(1+\rho)}{2}$, and $\frac{q_{1O}r_1(1+\rho)}{2}$, respectively. Calculating the conditional probabilities $p_x$ derived via the Bayes rule, the expected value of $u_S$ is then
\begin{equation}
u_S(1,1)=1-\frac{\beta_0 q_{O1}(1-q_{S2})(1-r_1)(1+\rho)}{\beta_1 q_{O1}(2-q_{S1}-q_{S1}\rho)+2\beta_0(1-q_{S2})}.
\label{us11}
\end{equation}
unless $q_{O1}=0$ and $q_{S2}=1$, and $u_S(1,1)=1$ otherwise.
%This is true for any off-equilibrium-path beliefs

{\em Step 2.} Let $x=1$ and suppose $y_S=2$. Then the possible combinations of $(y_S,y_O,d_S,d_O)$ and their respective probabilities, conditional on $b_S=1$, $x=1$, and $y_S=2$, are given by 

\begin{tabular}{cccc|c}
$y_S$&$y_O$&$d_S$&$d_O$&$p$\\
\hline
2&1&0&0&$\frac{1-r_1}{2}(2-q_{O1}-q_{O1}\rho)$\\
2&1&1&0&$\frac{r_1}{2}(2-q_{O1}-q_{O1}\rho)$\\
2&0&0&0&$\frac{1+\rho}{2}q_{O1}(1-r_1)^2$\\
2&0&0&1&$\frac{1+\rho}{2}q_{O1}r_1(1-r_1)$\\
2&0&1&0&$\frac{1+\rho}{2}q_{O1}r_1(1-r_1)$\\
2&0&1&1&$\frac{1+\rho}{2}q_{O1}r_1^2$
\end{tabular}
\vspace{.5cm}

Unless $q_{O2}=1$ and $q_{S1}=0$, the expected value of $u_S(1,2)$ is then as follows:  
\begin{eqnarray}
u_S(1,2)&=&\frac{1-r_1}{2}(2-q_{O1}-q_{O1}\rho)\frac{\beta_1 q_{S1}(2-q_{O1}-q_{O1}\rho)+2\alpha\beta_2(1-q_{O2})}{\beta_1 q_{S1}(2-q_{O1}-q_{O1}\rho)+2\beta_2(1-q_{O2})}+\frac{r_1}{2}(2-q_{O1}-q_{O1}\rho)+\nonumber\\
&+&\frac{1+\rho}{2}q_{O1}(1-r_1)^2\frac{\beta_1(1+\rho)q_{S1}q_{O1}(1-r_1)^2+2\alpha \beta_2q_{O2}(1-r_2)}{\beta_1(1+\rho)q_{S1}q_{O1}(1-r_1)^2+2(\beta_0q_{S2}+\beta_2q_{O2})(1-r_2)}+\nonumber\\
&+&\frac{1+\rho}{2}q_{O1}r_1(1-r_1)\frac{\beta_1(1+\rho)q_{S1}q_{O1}r_1(1-r_1)+2\alpha\beta_2q_{O2}r_2}{\beta_1(1+\rho)q_{S1}q_{O1}r_1(1-r_1)+2\beta_2q_{O2}r_2}+\nonumber\\
&+&\frac{1+\rho}{2}q_{O1}r_1(1-r_1)\frac{\beta_1(1+\rho)q_{S1}q_{O1}r_1(1-r_1)}{\beta_1(1+\rho)q_{S1}q_{O1}r_1(1-r_1)+2\beta_2q_{S2}r_2}+\frac{1+\rho}{2}q_{O1}r_1^2.
\label{us12}
\end{eqnarray}
If $q_{S1}=q_{O1}=0$ and $q_{S2}=q_{O2}<1$, then we have 
$u_S(1,2)=(1-r_1)\alpha+r_1(\pi_1+\alpha\pi_2)$, where $\pi_1$ and $\pi_2$ are beliefs about $x=1$ and $x=2$, respectively, in the off-path information set $y_S=2$, $y_O=1$, $d_S=1$, $d_O=0$. If the belief is a limit of a sequence of beliefs derived via the Bayes rule from a sequence of totally mixed strategy profiles converging to some $q_{S1}=q_{O1}=0$ and $q_{S2}=q_{O2}<1$, then we must have $\pi_1=1$ and $\pi_2=0$. Hence, $u_S(1,2)=(1-r_1)\alpha+r_1>u_S(1,1)$, and we cannot have $q_{S1}=q_{O1}=0$ and $q_{S2}=q_{O2}<1$ in an equilibrium. If $q_{S1}=q_{O1}=0$ and $q_{S2}=q_{O2}=1$, then $u_S(1,2)=(1-r_1)(\pi'_1+\pi'_2\alpha)+r_1(\pi_1+\alpha\pi_2)$, where $\pi'_1$ and $\pi'_2$ are beliefs about $x=1$ and $x=2$, respectively, in the off-path information set $y_S=2$, $y_O=1$, $d_S=1$, $d_O=0$. In any equilibrium we have $\pi'_1+\pi'_2=1$, so $u_S(1,2)=(1-r_1)(1-\pi_2'+\pi_2'\alpha)+r_1$ for some $\pi_2'\in[0,1]$. If we take $\pi'_1=1$, we than have $u_S(1,1)=u_S(1,2)$ at $q_{S1}=q_{O1}=0$ and $q_{S2}=q_{O2}=1$.

%However, let us impose trembling-hand perfection.... No, it will not work! If we assume that the belief is a limit of beliefs given $q_2\rightarrow 1$, $q_1\rightarrow 0$, and that these converge at the same rate, then we must have $\pi_2\in(0,1)$, so $u_S(1,2)>u_S(1,1)$ whenever $q_1=0$. It is always better to deviate. 

{\em Step 3.} Let $x=0$ and suppose $y_S=1$. Then, we invariably have $y_O=0$ and $d_O=0$. The possible combinations of $(y_S,y_O,d_S,d_O)$ and their respective probabilities, conditional on $b_S=1$, $x=0$, and $y_S=1$, are given by 

\begin{tabular}{cccc|c}
$y_S$&$y_O$&$d_S$&$d_O$&$p$\\
\hline
1&0&0&0&$1-r_1$\\
1&0&1&0&$r_1$
\end{tabular}
\vspace{.5cm}

\iffalse
Assuming that both players play strategies $(q_1,q_2)$, the probability $p$ of these values occurring in combinations with different values of $x$, and the conditional probabilities $p_x$ derived (whenever possible) by the Bayes rule, are given by the following table: 

\begin{tabular}{ccccc|cc}
$x$&$y_S$&$y_O$&$d_S$&$d_O$&$p$&$p_x$\\
\hline
0&1&0&0&0&$\frac{(1-\beta)(1-q_2)(1-r_1)}{4}$&$\frac{(1-\beta)(1-q_2)}{\beta q_1(2-q_1-q_1\rho)+(1-\beta)(1-q_2)}$\\
1&1&0&0&0&$\frac{\beta}{4}q_1(1-r_1)(2-q_1-q_1\rho)$&$\frac{\beta q_1(2-q_1-q_1\rho)}{\beta q_1(2-q_1-q_1\rho)+(1-\beta)(1-q_2)}$\\
2&1&0&0&0&0&0\\
0&1&0&1&0&$\frac{(1-\beta)(1-q_2)r_1}{4}$&1\\
1&1&0&1&0&0&0\\
2&1&0&1&0&0&0\\
\end{tabular}
\vspace{.5cm}
\fi
The information set $y_S=1$, $y_O=0$, $d_S=1$, $d_O=0$ lies off the game path if $q_{S2}=1$, but any sequence of totally mixed profiles converging to some $q_{S1}=q_{O1}=q_{S2}=q_{O2}=1$ will give us belief $p_0=1$, $p_1=p_2=0$ in that information set. This way, calculating the expected value gives us
\begin{equation}
u_S(0,1)=\frac{\beta_1 q_{O1}(1-r_1)(2-q_{S1}-q_{S1}\rho)}{\beta_1 q_{O1}(2-q_{S1}-q_{S1}\rho)+2\beta_0(1-q_{S2})}
\label{us01}
\end{equation}
unless $q_{O1}=0$ and $q_{S2}=1$. Otherwise, the payoffs depend on the beliefs in the information set $y_S=1$, $y_O=0$, $d_S=d_O=0$, with $u_S(0,1)=(1-r_1)\pi_1''$ for some $\pi_1''\in[0,1]$. 

{\em Step 4.} Let $x=0$ and $y_S=2$. The possible combinations of $(y_S,y_O,d_S,d_O)$ and their respective probabilities, conditional on $b_S=1$, $x=0$, and $y_S=2$, are given by 

\begin{tabular}{cccc|c}
$y_S$&$y_O$&$d_S$&$d_O$&$p$\\ 
\hline
2&0&0&0&$1-r_2$\\
2&0&1&0&$r_2$
\end{tabular}
\vspace{.5cm}

Calculating the expected value gives us
\begin{eqnarray}
u_S(0,2)&=&\frac{\beta_1(1+\rho)q_{S1}q_{O1}r_1(1-r_1)r_2}{\beta_1(1+\rho)q_{S1}q_{O1}r_1(1-r_1)+2\beta_0q_{S2}r_2}+\nonumber\\
&+&\frac{\beta_1(1+\rho)q_{S1}q_{O1}(1-r_1)^2(1-r_2)+2\alpha\beta_2q_{O2}(1-r_2)^2}{\beta_1(1+\rho)q_{S1}q_{O1}(1-r_1)^2+2(\beta_0q_{S2}+\beta_2q_{O2})(1-r_2)}.
\label{us02}
\end{eqnarray}
As established in Step 2, we cannot have $q_{S1}=q_{O1}=q_{S2}=q_{O2}=0$ in an equilibrium. If $q_{S1}=q_{O1}=0$ and $q_{S2}=q_{O2}=1$, then $u_S(0,1)\leq u_S(0,2)$ if $\pi_1''\leq\min\{1,\frac{\alpha}{2}\frac{1-r_2}{1-r_1}\}$. If $q_{S1}=q_{O1}>0$ and $q_{S2}=q_{O2}=0$, then $u_S(0,2)>u_S(0,1)$, so this cannot be an equilibrium. {\bf Q.E.D.}\\


{\bf Full proof of Proposition \ref{propq2}.} Fix $r_2=1$. Then $H_1(q_1,q_2)>0$, so $q_1=1$. We have $u_S(0,1)$ increasing in $q_2$ and $u_S(0,2)$ is decreasing in $q_2$, with $u_S(0,1)<u_S(0,2)$ at $q_2=0$. So, we either have $q_2=1$, or a closed-form solution for $u_S(0,1)=u_S(0,2)$ in $q_2$ exists, given by the expression (\ref{q2eq}). 
In order to do it we make several assumptions about beliefs in information sets that occur off the equilibrium paths. Recall that each information set is characterized by $(y_S,y_O,d_S,d_O)$, and consists of three elements $\{0,1,2\}$. 
First, we assume that zero probability is assigned to 2 if $y_S=0$ or to 0 if $y_O=2$. This means that only one-step backward misinformation is possible off the equilibrium path. Second, we assume that if $y_S$ and $y_O$ occur in equilibrium for some $x$ and do not occur for $x'\neq x$, then $\pi_{x'}=0$. This assumption implies that the observer believes in the possibility of a truthful report being mistakenly perceived as false. 

Let $x=1$ and $y_S=0$. Then the possible combinations of $(y_S,y_O,d_S,d_O)$ and their respective probabilities are given by 

\begin{tabular}{cccc|c}
$y_S$&$y_O$&$d_S$&$d_O$&$p$\\
\hline
0&0&0&0&$\frac{1+\rho}{2}q_1(1-r_1)^2$\\
0&0&1&0&$\frac{1+\rho}{2}q_1r_1(1-r_1)$\\
0&0&0&1&$\frac{1+\rho}{2}q_1r_1(1-r_1)$\\
0&0&1&1&$\frac{1+\rho}{2}q_1r_1^2$\\
0&1&1&0&$\frac{r_1}{2}(2-q_1-q_1\rho)$\\
0&1&0&0&$\frac{1-r_1}{2}(2-q_1-q_1\rho)$
\end{tabular}
\vspace{.5cm}

The first combination occurs in equilibrium only if $x=0$. The second, third, and fourth combinations can occur if $x=0$ and there is a nonzero probability of a truthful report interpreted as false. The fifth and sixth combinations do not occur in equilibrium. Under the one-step backward misinformation assumption, we must have $\pi_2=0$. This means that $u_S(1,0)\leq 1-\frac{1+\rho}{2}q_1<u_S(1,1)$.

Now let $x=2$. We have $u_S(2,2)>\frac{1-\rho}{2}\alpha+\frac{1+\rho}{2}$. Let $y_S=1$. We have $P(y_O=2)=\frac{1-\rho}{2}$. Under the one-step backward misinformation assumption, we must have $\pi_0=0$. Otherwise, if $y_O\in\{0,1\}$, we have $\pi_2=0$ either because the combination $(y_S,y_O,d_S,d_O)$ occurs in equilibrium, or because of our limiting assumptions on the off-the-path beliefs. Thus $u_S(2,1)\leq\frac{1-\rho}{2}\alpha+\frac{1+\rho}{2}<u_S(2,2)$. 

Finally, if $x=2$ and $y_S=0$. Then, the following combinations of $(y_S,y_O,d_S,d_O)$ are possible, with their probabilities:

\begin{tabular}{cccc|c}
$y_S$&$y_O$&$d_S$&$d_O$&$p$\\
\hline
0&2&1&0&$\frac{1-\rho}{2}$\\
0&1&1&0&$\frac{1+\rho}{2}(1-q_2)(1-r_1)$\\
0&1&1&1&$\frac{1+\rho}{2}(1-q_2)r_1$\\
0&0&1&1&$\frac{1+\rho}{2}q_2$\\
\end{tabular}

We have $\pi_1=1$ in the first case, $\pi_2=0$ in the second and third cases, and $\pi_0=1$ in the fourth case. This gives us $u_S(2,0)<1-q_2\frac{1+\rho}{2}$. 
{\bf Q.E.D.}\\

\section{Supplemental analysis: $r_2\in(r_1,1]$}

In this section, we consider the extension of the model to accommodate the more generic case where the probability of an egregiously false report being perceived as false can be less than one. Note that Lemmata \ref{lemma1}, \ref{lemma_pathological}, and \ref{lemma22} hold for all $r_2\in(r_1,1]$. Our results in the generic case will depend on the values of $\alpha$. 

\subsection{The case where $\alpha>2$}

The following statement gives the conditions for the existence of equilibrium $(1,1)$ if $\alpha>2$. In particular, it describes the different sets of parameter values $(r_1,r_2,\alpha,\beta)$ under which the existence of this equilibrium is guaranteed for all values of $\rho$, is conditional on $\rho$, or does not exist for any $\rho$: 

\begin{proposition}
\rm
Let $\alpha>2$. Put $\tilde r_2(r_1)=1-\frac{2(1-r_1)}{\alpha}$. There exist $0<r_1^{**}<r_1^{*}<1$ and increasing, differentiable functions $\bar r_2:[0,r_1^{*}]\rightarrow[0,1]$, ${\bar{\bar r}}_2:[r_1^{**},1]\rightarrow[0,1]$ satisfying
\begin{enumerate}
    \item $\bar r_2(r_1^{*})=\tilde r_2(r_1^*)$  and ${\bar{r}}_2(r_1)\in(r_1,\tilde r_2(r_1))$ for all $r_1<r_1^{*}$;
    \item ${\bar{\bar r}}_2(r_1^{**})=\tilde r_2(r_1^{**})$ and ${\bar{\bar r}}_2(r_1)\in(\tilde r_2(r_1),1]$ for all $r_1>r_1^{**}$.
\end{enumerate}
The existence of $(1,1)$ equilibrium depends on $r_1$, $r_2$, and $\rho$ in the following way:
\begin{enumerate}
    \item If $r_1\leq r_1^*$ and $r_2\in(r_1,\bar r_2(r_1)]$, or $r_1> r_1^*$ and $r_2\in(r_1,\tilde r_2(r_1)]$, then the equilibrium exists for all $\rho\in(-1,1)$; 
    \item If $r_1>r_1^{**}$ and $r_2\in(\tilde r_2(r_1),{\bar{\bar r}}_2(r_1))$, then the equilibrium exists if and only if $\rho\in[\rho^*,1)$ where $\rho^*\in(-1,1)$ is a value that depends on the parameters $r_1,r_2,\alpha,\beta$;
    \item If $r_1\leq r_1^{**}$ and $r_2\in[\tilde r_2(r_1),1]$, or $r_1> r_1^{**}$ and $r_2\in[{\bar{\bar r}}_2(r_1),1]$, then the equilibrium exists for no $\rho\in(-1,1)$; 
    \item If $r_1< r_1^*$ and $r_2\in(\bar r_2(r_1),\tilde r_2(r_1))$, then the equilibrium exists for some $\rho\in(-1,1)$ and does not exist for some other $\rho'\in(\rho,1)$. 
\end{enumerate}
\label{lemma_pureex}
\end{proposition}

The partition of the $(r_1,r_2)$ parameter space described in Proposition \ref{lemma_pureex} is illustrated in Figure \ref{eqex1}. 

\begin{figure}[ht]
\begin{center}
    \psfrag{r1}{$r_1$}
    \psfrag{r2}{$r_2$}
    \psfrag{a}{$r^{**}_1$}
    \psfrag{b}{$r^{*}_1$}
    \psfrag{A}{$A$}
    \psfrag{B}{$B$}
    \psfrag{C}{$C$}
    \psfrag{lab}{$\alpha=4,\beta=0.5$}
    \psfrag{c}{\rotatebox{25}{$\tilde r_2(r_1)$}}
    \psfrag{d}{\rotatebox{40}{$\bar r_2(r_1)$}}
    \psfrag{e}{\rotatebox{35}{${\bar{\bar r}}_2(r_1)$}}
    \includegraphics[width=4in]{cynic1.eps}
\end{center}


\caption{Existence of $(1,1)$ equilibrium depending on the values of $r_1$, $r_2$, and $\rho$, given $\alpha=4,\beta=0.5$. $A$ --- the equilibrium does not exists for any $\rho$; $B$ --- the equilibrium exists if $\rho$ is large enough; $C$ --- equilibrium exists for all $\rho$.}
\label{eqex1}
\end{figure}
%This is the value of $r_2$ for which $u_S(0,1)=u_S(0,2)$ as $r\rightarrow -1$. 

The existence of the pure-strategy equilibrium will be conditional on $\rho$ being large enough if two conditions on the other parameters are met. First, the probability $r_2$ has to be large enough relative to $r_1$. Second, $r_1$ cannot be too small; this corresponds to area $B$ on the graph. 

If the probability $r_2$ is not much higher than $r_1$, then a news outlet observing an unfavorable state of the world ($x=0$ for $S$ and $x=2$ for $O$) will choose an egregiously false report regardless of the value of $\rho$; this corresponds to region $C$ on the graph. 

Finally, if $r_2$ is large and $r_1$ is small (area $A$ on Figure \ref{eqex1}), always reporting $y_S=2$ if $x=0$ is not an equilibrium strategy, regardless of the cynicism parameter $\rho$. Suppose that the news outlets always make egregiously false reports. In that case, if an observer sees reports $y_S=2$, $y_O=0$, and knows that report $y_S$ is false, then it is likely that the state of the world is $x=0$, not $x=1$. Thus, if $r_1$ is low and $r_2$ is high, having one's egregiously false report observed as false does not seriously impact the reputation of the competing news outlet. 

We then proceed to investigate how parameters $\beta$ and $\alpha$ affect the existence of the pure-strategy equilibrium. 

\begin{proposition}
    \rm The following is true:
\begin{enumerate}[label=\roman*)]
    %\item We have $\frac{\partial r_1^{**}}{\partial\beta}<0$, $\lim_{\beta\rightarrow 1}r_1^{**}=0$, and $\lim_{\beta\rightarrow 0}r_1^{**}=r_1^{*}$; 
    %\item For every $r_1\in(r_1^{**},1)$ such that ${\bar{\bar r}}_2(r_1)<1$, we have $\frac{\partial {\bar{\bar r}}_2(r_1)}{\partial\beta}>0$, with $\lim_{\beta\rightarrow 0} {\bar{\bar r}}_2(r_1)=\tilde r_2(r_1)$;
    \item If $\beta>\frac{1}{3}$, then ${\bar{\bar r}}_2(r_1)=1$ if $r_1\in[\sqrt{\frac{1-\beta}{2\beta}},1]$;
    \item For every $(r_1,r_2,\beta,r)$, the pure-strategy equilibrium $(1,1)$ exists if $\alpha$ is large enough.
\end{enumerate}
\label{lemma2}    
\end{proposition}

The first part of this statement parallels Corollary \ref{corq2}; if $\beta$ increases, we should expect to see the $(1,1)$ equilibrium under a greater range of $(r_1,r_2)$. If $\beta$ is low, observed reports $y_S=2$ and $y_O=0$ will not likely result from the ruler being at normal strength. Hence, the value of making a report $y_S=2$ if the state of the world is $x=0$ will depend on both the probability $r_2$ of such a report being exposed as false and how cynical the audience is. We conclude the analysis of pure strategy equilibrium existence by looking at the effect of $\alpha$. When the state (opposition) media outlet is rewarded only if the public believes that the ruler is strong (weak), the outlets always choose to report their favorite states $y_S=2$ and $y_O=0$.

We next turn to investigate the existence and comparative statics of mixed-strategy equilibria. At least one equilibrium (mixed-strategy or pure-strategy) must exist, but for a generic $r_2$, there can be multiple equilibrium levels of $q_2$. Numeric simulation shows that, for some parameter values, there can exist either three mixed-strategy equilibria, two mixed-strategy equilibria and one pure-strategy equilibrium, or a unique equilibrium: either mixed-strategy or pure-strategy. Supplemental Appendix Figure \ref{eq_mult} shows the values for $r_1$ and $r_2$ for which multiple equilibria can exist, given some $\rho$. It appears that multiple equilibria are possible if $r_1$ is sufficiently small and $r_2$ is neither too large nor too small; also, if (given $r_1$ and $r_2$) the pure-strategy equilibrium exists for all values of $\rho$, then for no value of $\rho$ can one have multiple equilibria. 

While it is not possible to derive closed-form necessary and sufficient conditions for uniqueness of equilibria for the case of generic $r_2$, a sufficient condition guaranteeing uniqueness can be derived: 
\begin{lemma}
    \rm Suppose the following condition is satisfied:
\begin{equation}
r_1\geq\max\left\{\frac{1}{2},\frac{\alpha-2}{\alpha+2},\frac{\alpha-3}{\alpha-2}\right\}.
\label{r1bound}
\end{equation}
Then, the equilibrium with $q_1=1$ and $q_2>0$ is unique. 
\label{lemma_uniq}
\end{lemma}
Uniqueness is guaranteed if the probability that a moderately false report is exposed is large enough. However, if the value $\alpha$ is large, multiple equilibria may exist. 

Finally, we look at the comparative statics of mixed-strategy equilibrium for the generic case. %The expression for the sign of the derivative of $q_2$ with respect to $\rho$ will be nontransparent and unintuitive.
Condition (\ref{r1bound}) guarantees us that the comparative statics can be established:
\begin{proposition}
    \rm
    Suppose that (\ref{r1bound}) is satisfied and we have $q_2\in(0,1)$. Then we have $\frac{\partial q_2}{\partial \rho}>0$, $\frac{\partial q_2}{\partial \alpha}>0$, $\frac{\partial q_2}{\partial \beta}>0$, $\frac{\partial q_2}{\partial r_1}>0$, and $\frac{\partial q_2}{\partial r_2}<0$.
    \label{lemma_comp}
\end{proposition}

For $\beta$, $r_1$, and $\rho$, the direction of the effect is the same for generic $r_2$ as for the case where $r_2=1$.
Egregiously false statements are also made more frequently when $\alpha$ is higher and less frequently when the probability of an egregiously false statement being exposed is higher. 
 
%comparative statics result are parallels Propositions \ref{lemma_pureex} and \ref{lemma2} that describe the conditions for the existence of pure-strategy equilibria. 




\subsection{The case where $\alpha=2$}

This case is different from $\alpha>2$ because a continuum of equilibria, characterized by different levels of $q_1\in(0,1)$, may be possible. The following statement characterizes possible equilibria:
\begin{lemma}
\rm
Let $\alpha=2$. Then the following is true:
\begin{enumerate}[label=\roman*)]
\item If
$$
\frac{\beta}{1-\beta}(1+\rho)\geq\frac{r_2-r_1}{r_1(1-r_1)}
$$
and
$$
q_1\in\left[\sqrt{\frac{(1-\beta)(r_2-r_1)}{\beta (1+\rho)r_1(1-r_1)}},1\right],
$$
then $(q_1,1)$ is equilibrium;
\item There exists one equilibrium $(1,q_2)$ for some $q_2\in(0,1]$.
\end{enumerate}
\label{eq_al2}
\end{lemma}




The case $\alpha=2$ is special because we have $u_S(1,2)=u_S(1,1)$ whenever $q_2=1$, and $u_S(1,2)>u_S(1,1)$ otherwise. As a result, we have a continuum of equilibria for a nonzero measure set of $(\rho,r_1,r_2,\beta)$. If $q_2=1$, the only condition on $q_1$ is that $u_S(0,2)\geq u_S(0,1)$. Also, unlike in the case of $\alpha>2$ and a generic $r_2$, we are assured that there is only one equilibrium with $q_1=1$. 

Our next result characterizes the conditions for the existence of the pure-strategy equilibrium $(1,1)$. It is similar to Proposition \ref{lemma_pureex}, giving us the conditions on $r_1$ and $r_2$ under which this equilibrium can potentially exist. 

\begin{proposition}
\rm
Let $\alpha=2$. Then there exists an increasing, differentiable function ${\bar{\bar r}}_2:[0,1]\rightarrow[0,1]$ satisfying ${\bar{\bar r}}_2(0)=0$ and ${\bar{\bar r}}_2(r_1)>r_1$ for $r_1\in(0,1)$, and the existence of equilibrium with $q_1=q_2=1$ depends on $r_1$, $r_2$, and $\rho$ in the following way:
\begin{enumerate}
\item If $r_2\in(r_1,{\bar{\bar r}}_2(r_1))$, then the equilibrium exists if and only if $\rho\in[\rho^*,1)$ where $\rho^*\in(-1,1)$ is a value that depends on the parameters $r_1,r_2,\alpha,\beta$;
\item If $r_2\in[{\bar{\bar r}}_2(r_1),1]$, then the equilibrium exists for no $\rho\in(-1,1)$.
\end{enumerate}
\label{a2exist}
\end{proposition}

Unlike in the case $\alpha>2$, it is no longer possible for  equilibrium $(1,1)$ to exist for all values of $\rho\in(-1,1)$, given some $(r_1,r_2,\beta,\alpha)$. Either the pure-strategy equilibrium does not exist for all $\rho$, or for a high enough $\rho$. 

The comparative statics for the mixed-strategy equilibrium $(1,q_2)$, $q_2\in(0,1)$ are similar to ones for the case $\alpha>2$, but they are not limited by the condition (\ref{r1bound}): 
\begin{proposition}
    \rm
    Let $\alpha=2$ and suppose we have $q_2\in(0,1)$. Then we have $\frac{\partial q_2}{\partial \rho}>0$, $\frac{\partial q_2}{\partial \beta}>0$, $\frac{\partial q_2}{\partial r_1}>0$, and $\frac{\partial q_2}{\partial r_2}<0$. 
    \label{lemma_compstat2}
\end{proposition}




\subsection{The case where $\alpha<2$.}

This case differs from the previous two in that an egregiously false report is never made with probability one. Instead, it is possible to have a fully mixed equilibrium where maximal reports are chosen with probabilities strictly between zero and one:
\begin{lemma}
\rm Let $\alpha<2$. There exists an equilibrium $(q_1,q_2)$ with $q_1\in(0,1),\:q_2\in(0,1)$, and/or no more than one equilibrium $(1,q_2)$ with $q_2\in(0,1)$. An equilibrium $(q_1,1)$ does not exist if $q_1>0$.
\label{lemma221}
\end{lemma}



The equilibrium $(1,q_2)$ is guaranteed to exist if the cynicism parameter $\rho$ is sufficiently low. The probability of making an egregiously false report in this type of equilibrium invariably increases with $\rho$ and $\alpha$, and the comparative statics with respect to $r_2$ and $\beta$ are as in the $\alpha=2$ case:
\begin{proposition}
    \rm Let $\alpha<2$. Then for every $(r_1,r_2,\alpha,\beta)$ is exists $\check \rho\in(-1,1)$ such that for all $\rho\in(-1,\check \rho)$ there exists the $(1,q_2)$ equilibrium with $q_2\in(0,1)$. In that equilibrium we have $\frac{\partial q_2}{\partial\rho}>0$, $\frac{\partial q_2}{\partial \alpha}>0$, $\frac{\partial q_2}{\partial \beta}>0$, and $\frac{\partial q_2}{\partial r_2}<0$. 
    \label{rholow}
\end{proposition}


Proposition \ref{lemma221} does not rule out multiple equilibria $(q_1,q_2)$ with $q_1,q_2\in(0,1)$, or one such equilibrium in addition to $(1,q_2)$. While ruling out such multiple equilibria is not possible analytically, it can be examined using a numerical grid search algorithm. The set of equilibria was calculated for each $r_1\in\{0.05,0.1,\ldots, 0.95\}$, $r_2\in\{r_1+0.05,\ldots,0.95\}$, $\beta\in\{0.05,0.1,\ldots,0.95\}$, $\alpha\in\{1.05,\ldots,1.95\}$, $\rho\in\{-0.95,-0.9,\ldots,0.95\}$. Numeric analysis yielded two findings. First, there is a unique equilibrium distinct from $(0,1)$. Second, the cutoff value of $\rho$ from Proposition \ref{rholow} forms both a necessary and sufficient condition for having $q_2=1$ in equilibrium. 

%As $\beta\rightarrow 1$, we have $q_1\rightarrow 0$, $q_2\rightarrow 1$.

%$q_2$ monotonically decreases with $\rho$

%$q_1$ monotonically decreases with $\rho$
%Comp statics for $q_1\in(0,1)$, $q_2\in(0,1)$: can be complicated

\clearpage

\section*{Proofs of Appendix statements.}



{\bf Proof of Proposition \ref{lemma_pureex}.} Letting $R=1+\rho$, the condition
\begin{equation}
H_2(1,1)=\frac{\beta Rr_1(1-r_1)r_2}{\beta Rr_1(1-r_1)+(1-\beta)r_2}+\frac{\beta R(1-r_1)^2(1-r_2)+\alpha(1-\beta)(1-r_2)^2}{\beta R(1-r_1)^2+2(1-\beta)(1-r_2)}-(1-r_1)\geq 0
\label{HHH}
\end{equation}
is identical to
$$
AR^2+BR+C\geq 0,
$$
where
$$
A=\beta^2(1-r_1)^3r_1^2,\qquad B=\beta(1-\beta)(1-r_1)(r_1(5r_2+\alpha-2-r_2^2-3r_1r_2+2r_1+\alpha r_2^2-2\alpha r_2)-r_2^2),
$$
and
$$
C=(1-\beta)^2r_2(1-r_2)(\alpha(1-r_2)-2(1-r_1)).
$$
Also, denote $D=B^2-4AC$ and let $R_1\leq R_2$ be the (possible) real roots in $\rho$ of $AR^2+BR+C=0$. As the condition $u_S(0,2)=u_S(0,1)$ is tantamount to a quadratic equation in $\rho$ with a positive coefficient on $R^2$, it is sufficient to consider the following cases:

{\em Case 1.} $C\leq 0$ and $R_2\geq 2$. In that case, we have $u_S(0,2)>u_S(0,1)$ for all $\rho\in(-1,1)$. Condition $C\leq 0$ is identical to $r_2\geq\tilde r_2(r_1)$, while condition $R_1\geq 2$ translates to 
$$
G(r_1,r_2,\alpha,\beta)=4A+2B+C\geq 0.
$$
We have 
\begin{equation}
G(r_1,r_2,\alpha,\beta)=2\beta(1-r_1)^2\left(2\beta (1-r_1)r_1^2+ (1-\beta)(r_1^2(2\alpha+4)+r_1(\alpha^2-2\alpha)-(\alpha-2)^2\right).
\label{ggg}
\end{equation}
Note that $G(0,\tilde r_2(0),\alpha,\beta)<0$, $G(1,\tilde r_2(1),\alpha,\beta)=0$, $\frac{\partial G(1,\tilde r_2(1),\alpha,\beta)}{\partial r_1}>0$ at $r_1=0$, and that $G(r_1,\tilde r_2(r_1),\alpha,\beta)$ has a single critical point in $r_1$ on $[0,1)$ that is also a maximum. It follows that there is a single $r_1^{**}$ such that $C=4A+2B+C=0$. \\

{\em Case 2.} $C\leq 0$ and $R_2<2$. In that case, there exists a unique $r^*\in(-1,1)$ such that $u_S(0,2)=u_S(0,1)$. As we must have $G(r_1,r_2,\alpha,\beta)>0$, this is possible if and only if $r_1>r_1^{**}$ and $r_2\in(\tilde r_2(r_1),{\bar{\bar r}}_2(r_1))$.\\

{\em Case 3.} $C>0$ and either $R_1\geq 2$, $R_2<0$, or $D>0$. In that case, there does not exist $\rho\in(-1,1)$ such that $u_S(0,2)<u_S(0,1)$, so $q_1=q_2=1$ is equilibrium for all $\rho\in(-1,1)$. 

The unique solution in $r_1$ to $B=C=0$ is
\begin{equation}
r_1^{*}=\frac{\alpha-2}{\alpha+2}.
\end{equation}
Note that we have $1>r_1^{*}>r_1^{**}$ as $A>0$. If $r_1>r_1^{*}$ and $r_2\in(r_1,\tilde r_2(r_1))$, then we must have $C>0$ and $B>0$, so either $R_2<0$ or $D<0$, meaning that for no $\rho\in(-1,1)$ we have $u_S(0,2)=u_S(0,1)$. 

Note that $B$ decreases in $r_2$ over $[r_1,1]$ and $C$ increases in $r_2$ over $[r_1,\tilde r_2(r_1)]$. So for every $r_1<r_1^{*}$ there exists $\bar r_2(r_1)<\tilde r_2(r_1)$ such that $B<0$ and $D>0$ for every $r_2\in(\bar r_2(r_1),\tilde r_2(r_1)]$ and either $D<0$ or $B>0$ if $r_2<\bar r_2(r_1)$. Moreover, we have $\lim_{r_1\rightarrow r_1^{*}}\bar r_2(r_1)=\tilde r_2(r_1^{*})$. 
{\bf Q.E.D.}\\

{\bf Proof of Proposition \ref{lemma2}.} 
Condition $G(r_1,\tilde r_2(r_1),\alpha,\beta)=0$ for $r_1\in(0,1)$ is identical to 
$$
H_G=2\beta (1-r_1)r_1^2+ (1-\beta)(r_1^2(2\alpha+4)+r_1(\alpha^2-2\alpha)-(\alpha-2)^2.
$$
We have 
$$
\frac{\partial H_G}{\partial \beta}=(1-r_1)r_1^2-((r_1^2(2\alpha+4)+r_1(\alpha^2-2\alpha)-(\alpha-2)^2).
$$
This value is negative at $r_1=r_1^{**}$ because the value $(r_1^2(2\alpha+4)+r_1(\alpha^2-2\alpha)-(\alpha-2)^2)$ is negative at $r_1=r_1^{**}$, as $r_1^{**}<r_1^{*}$. Since $\frac{\partial H_G}{\partial r_1}>0$ at $r_1=r_1^{**}$, we have $\frac{\partial r_1^{**}}{\partial\beta}<0$. Moreover, if $\beta=1$, condition $H_G=0$ becomes identical to $(1-r_1)r_1^2=0$ so $r_1^{**}\rightarrow 0$ as $\beta\rightarrow 1$. If $\beta=0$, the condition becomes $(r_1^2(2\alpha+4)+r_1(\alpha^2-2\alpha)-(\alpha-2)^2)=0$ so $r_1^{**}=r_1^{*}$.

Now we have
$$
\frac{\partial H_2(1,1)}{\partial\beta}=\frac{1}{\beta(1-\beta)}\frac{\partial H_2(1,1)}{\partial \rho}.
$$
This value is positive as long as $r_1>r_1^{**}$ and $r_2\in(\tilde r_2(r_1),{\bar{\bar r}}_2(r_1))$. Now take $r_2=1$. The condition (\ref{HHH}) transforms to
$\beta(1+\rho)r_1^2\geq 1-\beta$. Let $\beta>\frac{1}{3}$. Then, this condition is satisfied as a strict inequality for all $\rho\in(0,1]$ if $r_1>\sqrt{\frac{1-\beta}{2\beta}}$ and as an equality if $r_1=\sqrt{\frac{1-\beta}{2\beta}}$. 

We have $\tilde r_2(r_1)=1-\frac{2(1-r_1)}{\alpha}$, so it approaches 1 as $\alpha$ becomes large. Now, as $\alpha\rightarrow \infty$, we have 
$$
4A+2B+C\approx \alpha(1-r_2)^2(\beta(1-\beta)(1-r_1)r_1+(1-\beta)^2r_2).
$$
The right-hand side of this expression is zero only if $r_2=1$. Hence, condition $4A+2B+C=0$ is satisfied for $r_2=1$ for the asymptotically large $\alpha$. Similarly, the condition $D=0$ is satisfied at $r_2=1$ for the asymptotically large $\alpha$. It follows that for every $r_1$ we have $\lim_{\alpha\rightarrow\infty}\bar r_2(r_1)=1$. {Q.E.D.}\\

{\bf Proof of Lemma \ref{lemma_uniq}.} In a mixed-strategy equilibrium, $q_2\in(0,1)$ must satisfy $H_2(1,q_2)=0$. Differentiating with respect to $q_2$ gives us
\begin{eqnarray}
\frac{\partial H_2(1,q_2)}{\partial q_2}&=&-\frac{\beta(1-\beta)Rr_1(1-r_1)r_2^2}{(\beta Rr_1(1-r_1)+(1-\beta)q_2r_2)^2}+\frac{\beta(1-\beta)Rq_2(1-r_1)^2(1-r_2)^2(\alpha-2)}{((\beta R(1-r_1)^2+2(1-\beta)q_2(1-r_2))^2}-\nonumber\\
&-&\frac{\beta(1-\beta)(2-R)(1-r_1)(1-q_2)}{(\beta (2-R)+(1-\beta)(1-q_2))^2}.
\label{dH2dq2}
\end{eqnarray}
It is true that 
\begin{eqnarray}
    \frac{\partial H_2(1,q_2)}{\partial q_2}&\geq& \beta(1-\beta)R(1-r_1)\left(\beta^2R^2(1-r_1)^3r_1((1-r_1)r_2^2-r_1^2(1-r_2)^2(\alpha-2))+\right.\nonumber\\
    &+&2\beta(1-\beta)Rq_2(1-r_1)^2(1-r_2)(2-r_1r_2(1-r_2)(\alpha-2))+\nonumber\\
    &+&\left.(1-\beta)^2q_2^2r_2^2(1-r_2)^2(4r_1-(1-r_1)(\alpha-2))\right).
    \label{HRR}
\end{eqnarray}
Since $r_1\leq r_2$, this condition transforms to (\ref{r1bound}). Now, as $H_2(1,0)>0$, it follows that either there is a unique mixed-strategy equilibrium if $H_1(1,1)<0$ or a unique pure-strategy equilibrium otherwise. {\bf Q.E.D.}\\


{\bf Proof of Proposition \ref{lemma_comp}.} Differentiating with respect to $R$ gives us
\begin{eqnarray}
\frac{\partial H_2(1,q_2)}{\partial R}&=&\frac{\beta(1-\beta)q_2r_1(1-r_1)r_2^2}{(\beta Rr_1(1-r_1)+(1-\beta)q_2r_2)^2}+\frac{\beta(1-\beta)q_2(1-r_1)^2(1-r_2)^2(2-\alpha)}{(\beta R(1-r_1)^2+2(1-\beta)q_2(1-r_2))^2}+\nonumber\\
&+&\frac{\beta(1-\beta)(1-r_1)(1-q_2)}{(\beta (2-R)+(1-\beta)(1-q_2))^2}.
\label{dH2dR}
\end{eqnarray}
The sufficient condition for the sum of the first and second parts of this expression to be positive is similar to (\ref{HRR}), up to a multiplication by $\frac{q_2}{R}$. 
The sign of $\frac{\partial q_2}{\partial \rho}$ follows from the implicit function theorem. Same for the sign of $\frac{\partial q_2}{\partial \beta}$, as $\frac{\partial H_2(1,q_2)}{\partial \beta}=\frac{R}{\beta(1-\beta)}\frac{\partial H_2(1,q_2)}{\partial \rho}$. 
Denote $X_1=\beta R(1-r_1)$, $X_2=(1-\beta)q_2$. Then we rewrite
$$
H_2(1,q_2)=\frac{X_1r_1r_2}{X_1r_1+X_2r_2}-(1-r_2)\frac{X_1(1-r_1)+\alpha X_2(1-r_2)}{X_1(1-r_1)+2X_2(1-r_2)}+C
$$
where $C$ does not depend on $r_2$. We have
\begin{align*}
&\frac{\partial H_2(1,q_2)}{\partial r_2}=\frac{X_1^2r_1^2}{(X_1r_1+X_2r_2)^2}-\frac{X_1^2(1-r_1)^2+2\alpha X_2(1-r_2)(X_1(1-r_1)+X_2(1-r_2))}{(X_1(1-r_1)+2X_2(1-r_2))^2}=\\
&=\frac{1}{(X_1r_1+X_2r_2)^2(X_1(1-r_1)+2X_2(1-r_2))^2}\times(2X_1^3X_2r_1(1-r_1)((2-\alpha)r_1(1-r_2)-r_2(1-r_1))+\\
&+X_1^2X_2^2(2(2-\alpha)r_1^2(1-r_2)^2-(1-r_1)^2r_2^2-4\alpha r_1r_2(1-r_1)(1-r_2))+X_1X_2^3D_1+X_2^4D_2),
\end{align*}
where $D_1<0$ and $D_2<0$; this expression is negative for all $\alpha>1$. So, by implicit function theorem, $q_2$ decreases with $r_2$. 

Denote $X_1'=\beta R$. We have
$$
H_2(1,q_2)=\frac{X_1'r_1(1-r_1)r_2}{X_1'r_1(1-r_1)+X_2r_2}+\frac{X_1'(1-r_1)^2(1-r_2)+\alpha X_2(1-r_2)^2}{X_1'(1-r_1)^2+2X_2(1-r_2)}-\frac{\beta (1-r_1)(2-R)}{\beta (2-R)+(1-\beta)(1-q_2)}
$$
and
\begin{equation}
\frac{\partial H_2(1,q_2)}{\partial r_1}=\frac{X_1'X_2r_2^2(1-2r_1)}{(X_1'r_1(1-r_1)+X_2r_2)^2}+\frac{2X_1'X_2(\alpha-2)(1-r_1)(1-r_2)^2}{(X_1'(1-r_1)^2+2X_2(1-r_2))^2}+\frac{\beta (2-R)}{\beta (2-R)+(1-\beta)(1-q_2)}.
\end{equation}
This expression is positive as $H_2(1,q_2)=0$; so, $q_2$ increases with $r_1$. {\bf Q.E.D.}\\



{\bf Proof of Lemma \ref{eq_al2}.} If $\alpha=2$, we have $H_1(q_1,1)=0$ for any $q_1\in(0,1]$. If
$$
q_1\geq \sqrt{\frac{(1-\beta)(r_2-r_1)}{\beta (1+\rho)r_1(1-r_1)}},
$$
then we have $H_2(q_1,1)\geq 0$, so $(q_1,1)$ can be an equilibrium. If $q_1=1$ and $q_2>0$, then also $u_S(1,2)\geq u_S(1,1)$. The uniqueness of such equilibrium follows from the fact that (\ref{dH2dq2}) is strictly negative, so we either have $H_2(1,q_2)=0$ for a single $q_2$, or $H_2(1,q_2)>0$ for all $q_2\in(0,1]$. {\bf Q.E.D.}\\


{\bf Proof of Proposition \ref{a2exist}.} This is a limit case of Proposition \ref{lemma_pureex} as $\alpha\rightarrow 2$. The unique solution to (\ref{ggg}) on $[0,1)$ is $r_1^{**}=0$. {\bf Q.E.D.}\\


{\bf Proof of Proposition \ref{lemma_compstat2}.} The signs of the derivatives are obtained in the proof of Proposition \ref{lemma_comp} and the fact that the sign of (\ref{dH2dq2}) is negative and the sign of (\ref{dH2dR}) is positive if $\alpha\leq 2$. {\bf Q.E.D.}\\

{\bf Proof of Lemma \ref{lemma221}.} Recall that $H_2$ strictly decreases with $q_2$ if $\alpha\leq2$ and that $H_2(q_1,0)>0$. Put $q^2_2(q_1)=\min\{q_2:H_2(q_1,q_2)=0,1\}$. This is a continuous function of $q_1$. Note also that $H_1$ strictly decreases with $q_2$. As $H_1>0$ when $q_2=0$ and $H_1<0$ when $q_2=1$, for each $q_1\in(0,1]$ there exists a unique $q^1_2(q_1)$ such that $H_1$ is satisfied and, moreover, $q_2^1$ is differentiable in $q_1$. Any equilibrium $(q_1,q_2)$ must satisfy $q_2=q^1_2(q_1)=q^2_2(q_1)$ if $q_1\in(0,1)$ and $q_2=q^2_2(1)\leq q^1_2(1)$ if $q_1=1$. As $q^1_2(q_1)<1$ we cannot have $q_2=1$ in equilibrium. To show that there exists an equilibrium $(q_1,q_2)$ with $q_1,q_2\in(0,1)$ if $q^2_2(1)\leq q^1_2(1)$, we linearize $q_2=1-xq_1$ in the neighborhood of $q_1=0$. Solving $H_2(q_1,1-xq_1)=0$ for $x$ we get $x=\frac{1}{1-\beta}(\frac{4\beta}{\alpha}\frac{1-r_1}{1-r_2}-2\beta)>0$. Solving $H_1(q_1,1-yq_1)=0$ for $y$ we obtain $y=0$, so $0=q^{1'}_2(0)>q^{2'}_2(0)=-x$, and there must exist $q_1\in(0,1)$ such that $q^1_2(q_1)=q^2_2(q_1)$. {\bf Q.E.D.}\\

{\bf Proof of Proposition \ref{rholow}.} Recall that $H_1$ and $H_2$ decrease in $q_2$. We have equilibrium $(1,q_2)$ with $q_2\in(0,1)$ if $H_2(1,q_2)=0$ and $q_2\leq\max\{q_2'|H_1(1,q_2')\geq 0\}$. As $\rho\rightarrow-1$, the solution to $H_1(1,q_2)$ in $q_2$ approaches 1, and the solution to $H_2(1,q_2)$ in $q_2$ approaches $\max\{0,\frac{1+\beta}{1-\beta}-\frac{4\beta}{\alpha(1-\beta)}\frac{1-r_1}{1-r_2}\}<1$. It follows that $q_1=1$ and $q_2\in(0,1)$ if $\rho$ is small enough. The signs of the comparative statics derivatives are obtained in the proof of Proposition \ref{lemma_comp} and the fact that the sign of (\ref{dH2dq2}) is negative and the sign of (\ref{dH2dR}) is positive if $\alpha\leq 2$.  {\bf Q.E.D.}\\


\clearpage
\section{Supplemental figures}

\begin{figure}[ht]
\begin{center}
\subfigure[$\alpha=3$, $\beta=0.5$]{
    \psfrag{r1}{$r_1$}
    \psfrag{r2}{$r_2$}
    \psfrag{A}{$A$}
    \psfrag{B}{$B$}
    \psfrag{C}{$C$}    
\includegraphics[width=3in]{cyn5_30_mult.eps}
}
\subfigure[$\alpha=3$, $\beta=0.8$]{
\psfrag{r1}{$r_1$}
    \psfrag{r2}{$r_2$}
    \psfrag{A}{$A$}
    \psfrag{B}{$B$}
    \psfrag{C}{$C$}
\includegraphics[width=3in]{cyn8_30_mult.eps}
}
\subfigure[$\alpha=5$, $\beta=0.5$]{
\psfrag{r1}{$r_1$}
    \psfrag{r2}{$r_2$}
    \psfrag{A}{$A$}
    \psfrag{B}{$B$}
    \psfrag{C}{$C$}
\includegraphics[width=3in]{cyn5_50_mult.eps}
}
\subfigure[$\alpha=5$, $\beta=0.8$]{
\psfrag{r1}{$r_1$}
    \psfrag{r2}{$r_2$}
    \psfrag{A}{$A$}
    \psfrag{B}{$B$}
    \psfrag{C}{$C$}
\includegraphics[width=3in]{cyn8_50_mult.eps}
}
\caption{Multiple equilibria ($\Diamond$ - two mixed-strategy and one pure-strategy equilibrium possible for some $\rho$; $+$ - three mixed-strategy equilibria possible for some $\rho$).}
\label{eq_mult}
\end{center}
\end{figure}

\end{document}